Problem 81 - A Complete Resource Book in Mathematics for JEE Main

Question

If \(\lim _{x \rightarrow 0}\left[1+x+\frac{f(x)}{x}\right]^{1 / x}=e^{3}\), then \(\lim _{x \rightarrow 0}\left[1+\frac{f(x)}{x}\right]^{1 / x}=\) (A) \(e\) (B) \(e^{2}\) (C) \(e^{3}\) (D) None of these

Step-by-Step Solution

Answer

The limit evaluates to \(e^{2}\), which is option (B).

Step 1: Interpret the given limit

We are given that \( \lim_{x \to 0}\left[1+x+\frac{f(x)}{x}\right]^{1/x} = e^3 \). This means that as \( x \) approaches zero, the expression inside the limit approaches \( e^3 \).

Step 2: Break down the expression

Rewrite the expression: \( \left[1+x+\frac{f(x)}{x}\right]^{1/x} \). Notice this combines two parts: \(1+x\) and \(\frac{f(x)}{x}\). We'll focus on finding what \(\lim_{x \to 0}\left[1+\frac{f(x)}{x}\right]^{1/x}\) equals.

Step 3: Apply limit properties

Assume \( \lim_{x \to 0} \left[1+\frac{f(x)}{x}\right]^{1/x} = e^a \). Then, \( \lim_{x \to 0} \left[1+x+\frac{f(x)}{x}\right]^{1/x} = e^{a+1} \) due to the \(x\) added inside the exponent.

Step 4: Use the given conditions

We know \( e^{a+1} = e^3 \). Therefore, \( a+1 = 3 \), leading to \( a = 2 \).

Step 5: Conclude the evaluation

Finally, from \( a = 2 \), we find \( \lim_{x \to 0}\left[1+\frac{f(x)}{x}\right]^{1/x} = e^2 \).

Key Concepts

Exponential Functions

Exponential functions are a cornerstone of calculus, characterized by the constant base 'e' raised to the power of a variable exponent. They appear commonly when discussing growth and decay processes. In calculus, the natural exponential function is described as \(e^x\), where 'e' is approximately 2.718. It's unique because the rate of change of \(e^x\) at any point is equal to its value at that point.
Understanding how limits work in the context of exponential functions helps evaluate expressions that might seem complex initially. For example, when you have limits like \(\lim_{x \to 0} [1 + x + \frac{f(x)}{x}]^{1/x} = e^3\), it indicates how the behavior of the function changes as 'x' approaches 0.
In this problem, recognizing the format \([1 + f(x)]^{1/x}\) hints at limits commonly associated with exponential behavior. By manipulating the form within the brackets, scholars can simplify complex expressions into an exponential function involving 'e', ultimately leading to insights about the behavior of the function as \(x\) nears specific values like 0.

Limit Properties

The properties of limits form the backbone of evaluating complex expressions and understanding their behavior under certain conditions. When handling indeterminate forms or functions that aren't initially straightforward, limit properties such as multiplication, division, and addition rules prove invaluable.
One crucial property is the limit of a sum: the limit of the sum of two functions is the sum of their limits if both limits exist. This is pivotal when breaking down expressions like \(1+x+\frac{f(x)}{x}\). As the solution shows, decomposing the original expression into parts—focusing squarely on separate components—allows more manageable manipulation.
Additionally, when the limit approaches an exponent, like \(e\), multiplying and adjusting limits based on behavior 'inside' the brackets is key. For instance, translating \([1 + x + \frac{f(x)}{x}]^{1/x}\) into \([1+\frac{f(x)}{x}]^{1/x}\) accompanied by a small adjustment yields the fundamental limit statement: \(e^{a+1} = e^3\), leading to solving \(a = 2\). It showcases how a clear understanding of limit properties expedites finding solutions.

Indeterminate Forms

When dealing with limits, you'll often encounter indeterminate forms like \(\frac{0}{0}\) or \(\frac{∞}{∞}\). They arise in scenarios where direct substitution within a limit doesn't provide useful information. Instead, they necessitate algebraic or calculus-based manipulation to uncover the actual limit value.
In this exercise, indeterminate forms are subtly present when \(x\) approaches zero. The expression \(\left[1+x+\frac{f(x)}{x}\right]^{1/x}\) becomes convoluted directly, thus initially seeming to represent an indeterminate form like \(e^0 = 1\) without any information on growth or decline behavior from the function \(f(x)\).
The strategic approach—recognizing outside additions inside a limit and dissecting terms into manageable pieces—avoids pitfalls of directly grappling with indeterminate forms. As we solve, transforming expressions into a format conducive for limit calculation, such as refining to \([1+\frac{f(x)}{x}]^{1/x}\), removes ambiguity and reveals the precise behavior as \(x\) approaches the critical point. Conclusively, understanding indeterminate forms and overcoming them is essential in dissecting complex calculus problems.